Influence of turbulence on the dynamo threshold

1. Influence of turbulence on the dynamo threshold B. Dubrulle, GIT/SPEC N. Leprovost, R. Dolganov, P. Blaineau J-P. Laval and F. Daviaud

2. Basic equations Maxwell equations Navier-Stokes equations Field strectching Field diffusion Competition characterized by magnetic Reynolds number (Instability) Dynamo if Rm > Rmc

3. « Classical dynamo » paradigm Em Em t t Rm Dynamo Pas dynamo Rmc Indicator: Dynamo No dynamo

4. Dynamos in the Universe Def: Magnetic field generation through movement of a conductor In the universe…. stars, galaxies planets Control Parameters:

5. Problem Turbulent flow: Fluctuation Mean Flow Multiplicative noise Classical linear instability 

6. « Classical turbulence » paradigm Mean Field argument: Mean Field equation: Mean Field dispersion: Mean Field instability: Turbulence creates dynamo « most of the time » « Helical turbulence is good for dynamo »

7. Numerical test ? Whithout mean velocity With mean-velocity Pm=1 Pm=1 Rm Re Ponty et al, 2005, 2006 Laval et al, 2006 Schekochihin et al, 2004

8. Experimental test ? *Karlsruhe

9. Dynamos with low “unstationarity”: success Riga Karlsruhe A. Gailitis et al., Phys. Rev. Lett., 2001 R Stieglitz, U. Müller, Phys.of Fluids,2001

10. “TM60”, no dynamo Field “TM28”, dynamo Experiments with unstationarity… VKS Experiment Sodium Optimisation Measure Dynamo No dynamo Kinematic code

11. …Failure!

12. Explanation: numerics Simulation with real velocity Turbulence increases threshold With respect to time-averaged! Simulation with time averaged velocity

13. Explanation: theory Mean Field Theorie (<V>=0) Perturbative computation (Petrelis, Fauve) (with mean velocity fiel) Kraichnan model (<V>=0) Dynamo only for Who is right ??????

14. Importance of the order parameter MFT, Petrelis, Fauve: transition over <B> KM : transition over <B2> <B>=0… No dynamo (MFT, Petrelis) <B2> non zero…Dynamo (KM) Vote: What is good order parameter?

15. Problem Turbulent flow: Fluctuation Mean Flow Multiplicative noise Classical linear instability 

16. Troubles Model equation: Problem: how to define threshold? Etc, etc... Instability threshold depends on moment order!!! Solution: work with PDF and Lyapunov exponent

17. Stochastic approach Basic Equations Approximation 1 Approximation 2 Mean Flow Noise delta-correlated in time

18. Fokker-Planck Equation Equation for P(B,x,t) with

19. Mean-Field Equation beta effect (turbulent diffusion) Alpha effect Helicity if isotropy Mean Field Theory Equation Stability governed by alpha et beta….!!!???

20. Isotropic case Mean Field Magnetic energy

21. Stationary solutions Always solution! Other solution: Z: normalization D: space dimension a et : coefficients Lyapunov exponent!

22. Bifurcations Non-zero Solution (normalisation) Most probable value Intermittent Dynamo Turbulent Dynamo  No dynamo 0 aD

23. New theoretical turbulent paradigm Laminar Em Em t t Rm Dynamo Pas dynamo Rmc Turbulent Intermittent Dynamo Turbulent Dynamo Rm No dynamo Rm1 Rm2

24. The Lyapunov exponent… >0 and proportional to noise ( KM effect) Orientation (<0) (zero if <V>=0) Unstable Direction Stable Direction Rmc Rmc <V> Expected result Noise intensity Leprovost, Dubrulle, EPJB 2005

25. Illustration: Bullard Homopolar Dynamo Intermittent Dynamo No Dynamo Noise intensity Leprovost, PhD thesis

26. Discussion Noise influences threshold through mu AND vector orientation Turbulent threshold can be very different from « laminar » ones Influence of alpha and beta through vector orientation Threshold different from Mean Field Theory prediction Dangerous to optimize dynamo experiments from mean field!

27. Simulations Time-average of velocity field computed through Navier-Stokes Type of simulation MHD-DNS Kinematic Noisy Computed through NS 0 Markovian noise (F,tc, ki)

28. Numerical code Spectral method Integration scheme: Adams-Brashford Resolution: 64*64*64 to 256*256*256 Forcing with Taylor-Green vortex Constant velocity forcing Cf Ponty et al, 2004, 2005

29. Time-averaged vs real dynamo • Laval, Blaineau, Leprovost, Dubrulle, FD: PRL 2006 2 dynamo windows

30. Results for noisy delta-correlated Forcing at ki=1 Forcing at ki=16 Linear in (-1) (Fauve-Petrelis)

31. Results for noisy tc=0.1 Forcing at ki=16 Forcing at ki=1

32. Results for noisy tc=1 s Forcing at ki=1

33. Summary of noisy 50 Compa DNS Stochastic noise k=1 Tauc=1 s ki=1 8 1 0 0.1 ki=16 DNS Summary of noisy simulations Tc=1

34. Interpretation Rm   Universal curve Rm* Kinetic energy of of the Velocity Fluid

35. Definition of a universal « control parameter » In VKS =30 =0.97

36. Comparison stochastic/DNS Tauc Compa DNS Stochastic noise k=1 Tauc=1 s ki=1 ki=16 Summary of noisy simulations

37. Comparison DNS and mean flow Dynamo CM Intermittent Dynamo No dynamo Laval, Blaineau, Leprovost, Dubrulle, Daviaud (2005)

38. Conclusions In Taylor-Green, turbulence is not favourable to dynamo Turbulence looks like a large scale noise Bad influence through desorientation effect Large scale turbulence (unstationarity) increases dynamo threshold -> desorientation effect Small scale turbulence decreases dynamo threshold-> « friction » Possible transition to dynamo via intermittent scenario In natural objects: importance of Coriolis force (kills large-scale) Possibility of stochastic simulations to replace DNS